Demand shaping in an electrical power grid

ABSTRACT

A method ( 100 ) for demand shaping through load shedding and shifting in an electrical smart grid.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of Provisional U.S. PatentApplication Ser. No. 61/773,996, entitled “Robust Demand Shaping throughLoad Shedding and Load Shifting with a Two-Level Market”, filed in thename of Rodrigo Carrasco, Ioannis Akrotirianakis and Amit Chakraborty onMar. 7, 2013, the disclosure of which is also hereby incorporated hereinby reference.

FIELD OF INVENTION

The present invention relates to the control of electricity flows in anelectrical power network. More particularly, the present inventionrelates to the control of smart electrical power networks (or grids).

BACKGROUND OF THE INVENTION

The electrical power grid is experiencing important changes. First,introducing new power plants and enlarging the grid is becoming moreexpensive and complicated. Second, the introduction of more renewablepower sources, which have a much higher output variance, will make iteven harder to control and predict the state of the system. In thisenvironment, a smart grid, where not only energy but also data istransmitted, appears to be a solution in the right direction as it givesfurther control to all parties involved in the various aspects of thepower grid: generation, transmission, distribution, and consumption.Smart grids are the evolution of our current electrical grid and promiseto solve many of the current limitations like managing sources withhigher variability and increase security and reliability. But in orderto use smart grids appropriately, new mechanisms and processes need tobe put in place to control the flow of energy and achieve the promisedgoals. Demand shaping mechanisms are among the most crucial ones, sincethey allow the grid operators to control and shape the demand, reducecosts and peak to average consumption, as well as increase reliabilityand better control blackouts and brownouts.

The current smart grid approaches and related work may be organizedgenerally into two main categories: demand response (DR) methods andgrid/market modeling. Several different methods for demand response havebeen proposed in the literature. A good summary of the different DRapproaches is presented in an article by M. Albadi and E. El-Saadanyentitled “Demand response in electricity markets: An overview”, PowerEngineering Society (2007), pp. 1-5 and in their follow up articleentitled “A summary of demand response in electricity markets”, ElectricPower Systems Research 78, No. 11 (November 2008), pp. 1989-1996. Anadditional, interesting source of DR approach results can be found in anarticle by J.-H. Kim and A. Shcherbakova entitled “Common failures ofdemand response”, Energy 36, No. 2 (February 2011), pp. 873-880, inwhich the authors list several implementations and trials done as wellas the shortcomings they have observed, shedding light into whichmethods are better for controlling demand level.

The simplest approach of demand response methods is the direct control,which basically uses the ideas of dynamic demand implemented in GreatBritain. An elaborate example of that approach appears in an article byG. Tejeda and A. Cipriano entitled “Direct Load Control of HVAC systemsusing Hybrid Model”, Predictive Control (2012), in which the authorscontrol a HVAC (heating, ventilation and air conditioning) loaddirectly. Their approach comes from the control theory perspective but,if the objective functions are changed for other economic quantities, acontrol system could be implemented that minimizes a desired metric,like generation cost or peak to average ratio (PAR), for electricalpower grid purposes.

In terms of price/incentive based mechanisms, there are three mainmethods in demand response. The first one focuses on demand shedding asshown in an article by L. Chen, N. Li, S. H. Low and J. C. Doyle,entitled “Two Market Models for Demand Response in Power Networks ”, In2010 First IEEE International Conference on Smart Grid Communications,(October 2010), pp. 397-402, IEEE. In this method, users will shed partof their demand, and the level of that shedding depends on the priceoffered by the generator or utility company. Thus, the results of theabove-referenced article point towards how to compute the equilibriumprices such that a known amount d of demand can be shed. The authorsconsider that users shed demand linearly with prices, which could betrue for small values of d but as d increases their model certainly willnot work.

The second method is demand-source balancing. This method makes surethat the demand adapts to the current generation levels, somethingespecially important in the presence of variable energy sources likewind or sun Several different methods are proposed in the literature(for example, in an article by A. D. Dominguez-Garcia and C. N.Hadjicostis entitled “Distributed algorithms for control of demandresponse and distributed energy resources”, In Decision and Control(2011), pp. 27-32; in an article by P. Loiseau, G. Schwartz and J.Musacchio, entitled “Congestion pricing using a raffle-based scheme”, InNetwork Games, Control and Optimization, No. 2, (2011) pp. 1-8; in anarticle by A.-H. Mohsenian-Rad, V. W. S. Wong, J. Jatskevich, R.Schober, and A. Leongarcia, entitled “Autonomous Demand-Side ManagementBased on Game-Theoretic Energy Consumption Scheduling for the FutureSmart Grid”, IEEE Transactions on Smart Grid 1, No. 3 (December 2010),pp. 320-331; and in an article by F. Partovl, M. Nikzad, B. Mozafari,and A. M. Ranjbar, entitled “A stochastic security approach to energyand spinning reserve scheduling considering demand response program”,Energy 36, No. 5 (May 2011), pp. 3130-3137). One of the most interestingmethods is the one proposed by the Mohsenian-Rad, et al. article abovesince the authors arrive to a distributed method that can balance thedemand level given the generation output with the objective ofminimizing the generation cost.

The third main demand response method is demand shifting. This is theone most directly related to the problem of demand shaping since in thiscase the users agree to shift their loads in time, according to a pricesignal by the utility company. An example of such an approach appears inan article by M. Kraning, E. Chu, J. Lavaei, S. Boyd, and W. D. April,entitled “Message Passing for Dynamic Network Energy Management” (2012)in which users give their preferences and restrictions for theirdifferent appliances (like washing machines, electric vehicles, etc.)and then, through a distributed algorithm, the demand level is definedto achieve a certain control objective. The main issue with this methodis that it is not truthful (i.e. the users can cheat the system by notrevealing their true requirements and thus obtain benefits above thesocial optimum). Also, there are some privacy concerns in terms of theinformation the users must share. Still, it is a very interestingapproach to see how loads can be shifted around.

More related to the methodology side, one article (by W. Chen, D. Huang,A. A. Kulkarni, J. Unnikrishnan, Q. Zhu, P. Mehta, S. Meyn, and A.Wierman entitled “Approximate dynamic programming using fluid anddiffusion approximations with applications to power management”,Proceedings of the 48th IEEE Conference on Decision and Control (CDC),(December 2009), pp. 3575-3580) discusses one methodology that can beused to compute optimal solutions of complex models. This method couldbe interesting if there is a need to find an equilibrium point or aprice value, and, especially, if there is a focus on using stochasticmodels to represent a system.

The literature also discusses how to model the power grid and moreimportantly, how to model or find solutions for the market that worksover the grid. It is important to remember that this market basicallyworks in three levels: a long-term market, day-ahead market, and realtime market. In the long term market generators and large consumers orutility companies sign agreements for power delivery many weeks, months,or even years ahead. Then in the day-ahead market utility companiespurchase whatever additional energy they might need given the muchbetter forecasts they have for the next day, as well as the reserverequired. The third market is the real time market or spot market, whichis between 5 to 10 minutes ahead of the actual real-time demand and itis used to match the demand exactly.

In one article, the authors highlight the difficulties the future gridwill have, which is important since some of the new metrics a futuresmart grid might be of interest while solving current issues with demandshaping (see the article by M. Negrete-Pincetic and S. Meyn, entitled“Intelligence by Design for the Entropic Grid”, pp. 1-8). One of themost interesting and useful models for the energy market is the onedeveloped in a document by L-K. Cho and S. P. Meyn, entitled “Efficiencyand marginal cost pricing in dynamic competitive markets with friction”,Theoretical Economics 5, No. 2 (2010), pp. 215-239 in which the authorsformulate the general model and are able to compute equilibrium pointsgiven certain simplifications. The main takeaway of this document isthat, by adding friction to the market model (which appears due to theramp-up constraints given by generators), it is now possible to achievesolutions with price volatility similar to the ones observed in realenergy markets. This is a key point since the first step in most otherpapers in the area is to simplify the model by eliminating ramp-uprequirements, and thus those results could be far off from reality. Infollow-up papers, the respective authors further analyze their modelswith additional components, such as variable energy sources (see thepapers by S. Meyn, M. Negrete-Pincetic, G. Wang, A. Kowli, and E.Hafieepoorfard entitled “The value of volatile resources in electricitymarkets”, 49th IEEE Conference on Decision and Control (CDC), (December2010), pp. 1029-1036; by G. Wang, A. Kowli, M. Negrete-Pincetic, E.Shafreepoorfard and S. Meyn entitled “A Control Theorist s Perspectiveon Dynamic Competitive Equilibria in Electricity Markets”, In I FAC,(2011); and by G. Wang, M. Negrete-Pincetic, A. Kowli, S. Meyn, and U.V. Shanbhag, entitled “Dynamic Competitive Equilibria in ElectricityMarkets”, In Control and Optimization Theory Jor Electric Smart Grids,2011, pp. 1-28). These are very interesting results since they are ableto model the system considering that both demand and generation ofenergy are stochastic processes, which is useful for other approaches.Another article analyzes the system in terms of its reliability, whichis something that might also be of interest (see the article by M. Chen,L-K. Cho, and S. P. Meyn, entitled “Reliability by design in distributedpower transmission networks”, Automatica 42, No. 8 (August 2006), pp.1267-1281).

Another approach for modeling this market is given in an article by S.Worgin, B. F. Hobbs, D. Ralph, E. Centeno, and J. Barquin, entitled“Open versus closed loop capacity equilibria in electricity marketsunder perfect and oligopolistic competition”, pp. 1-36, in which theauthors not only analyze the perfect competition setting but also theoligopolistic one in which energy producers are not price takers as inmost of the rest of the literature.

Finally, how to add variable sources to the energy market model can befound in an article by J. Nair, S. Adlakha, and A. Wierman, entitled“Energy Procurement Strategies in the Presence of Intermittent Sources”,2012. The most interesting result in this article is how they are ableto compute procurement strategies when the energy sources have highvariability while at the same time being able to model the three levelsof the energy market. They do it by adding error to the forecasts foreach of the different markets which could be useful for otherforecasting methodologies.

SUMMARY OF THE INVENTION

The above problems are obviated by the present invention which providesa method for controlling the flow of energy in an electrical power grid,comprising obtaining an optimal load shedding profile for a user of thegrid; obtaining an optimal billing structure and an optimal loadshifting profile for the user; iterating the steps of obtaining anoptimal load shedding profile and obtaining an optimal billing structureand optimal load shifting profile until convergence is achieved; andcontrolling the energy consumption of the user based on the optimal loadshedding profile, optimal billing structure, and optimal load shiftingprofile. The step of obtaining an optimal load shedding profile mayutilize a known billing structure and a known load shifting profile foreach user of the grid. In such case, the step of obtaining an optimalload shedding profile may be performed without express information ofload preferences of each user of the grid. Each of the steps ofobtaining an optimal load shedding profile and obtaining an optimalbilling structure and an optimal load shifting profile may be a solutionto a robust convex optimization problem.

The present invention also provides a method for demand shaping throughload shedding and load shifting in a electrical power smart grid,comprising directly controlling the load of a user to time shift part ofthe load demand of the user; providing energy purchase price incentivesto the user to control the level of shedding part of the load demand ofthe user; and determining how much energy to purchase on each of theday-ahead market and the real-time market to cover the load demand ofthe user after load shifting and load shedding. The step of directlycontrolling the load of a user may comprise calculating an optimal loadshifting profile of the user. In such case, the step of calculating anoptimal load shifting profile of the user may comprise utilizing arobust optimization problem that takes into account that a user loaddemand level and energy purchase prices are unknown stochasticprocesses. The step of providing energy price incentives to the user maycomprise calculating an optimal load shedding profile of the user. Insuch case, the step of calculating an optimal load shedding profile ofthe user may comprise utilizing a robust optimization problem that takesinto account user load demand level and energy purchase prices areunknown stochastic processes.

The present invention may also provide a method of controlling theenergy demand of a user of an electrical power network, comprisingdetermining an optimal amount of user energy demand to shed and a priceincentive required to achieve said optimal amount; determining a billingstructure and an optimal amount of user energy demand to time-shift;repeating the first and second method steps until an equilibrium isachieved; and controlling the energy demand of the user based thedetermined optimal amount of user energy demand to shed, price incentiverequired to achieve said optimal amount, billing structure, and optimalamount of user energy demand to time-shift. The step of determining theoptimal amount of user energy demand to shed may comprise formulatingand solving an optimization problem using a known billing structure anda known user consumption for shiftable loads. In such case, the step offormulating and solving may comprise utilizing standard convexoptimization techniques to solve the optimization problem in the casethat energy price incentives are set by a party other than a networkoperator. Alternatively, the step of formulating and solving maycomprise setting a price offer for the user and re-formulating andsolving the re-formulated optimization problem with user preferences toload shedding in the case that energy price incentives are set by anetwork operator. The step of determining a billing structure and anoptimal amount of user energy demand to time-shift may compriseformulating and solving an optimization problem based on results forcalculating an optimal amount of user energy demand to shed and a priceincentive required to achieve said optimal amount. In such case, thestep of formulating and solving may comprise formulating and solvingrespective billing structures for each of user shiftable plus sheddableload and user non-moveable load using a fixed utility ratio. Further,the step of formulating and solving respective billing structures maycomprise re-formulating and solving re-formulated optimization problemto obtain an optimal amount of user energy demand to time-shift.

The present invention may also provide a control system for anelectrical power network, comprising a controller that shapes theelectricity demand of users by determining electricity purchase pricesand billing structures in an iterative manner to achieve an equilibriumand by determining the allocation of electricity demand for each usergiven the determined electricity purchase prices and billing structures.

The present invention may also provide a smart electrical power networkthat provides electricity to users and communicates data between itsvarious components to control the flow of electricity, comprising acontrol system that is adapted to shape user electricity demand bysimultaneously shifting and shedding user electricity demand and byprocuring electricity provided to users from a two-stage market with aday-ahead market (DAM) and a real-time market (RTM).

Advantageously, the present invention provides a new mechanism to givefurther control of the power flow to utility companies, allowing them toshape the user demand in order to minimize the costs of procuring thatenergy.

DESCRIPTION OF THE DRAWINGS

For a better understanding of the present invention, reference is madeto the following description of an exemplary embodiment thereof, and tothe accompanying drawings, wherein:

FIG. 1 is a block diagram of a smart electrical power network(simplified);

FIG. 2 is a schematic representation of a control method implemented inaccordance with the present invention;

FIG. 3 a shows histograms of synthetic data of monthly demanddistribution for a user;

FIG. 3 b shows histograms of synthetic data of hourly demanddistribution for a user;

FIG. 3 c shows histograms of synthetic data of single day of the weekand single hour demand distributions for multiple users;

FIG. 3 d shows synthetic data demand distribution comparison (same dayof the week and hour) with empirical data for a user;

FIG. 3 e is a chart of simulated average user demand for each usercluster for one day;

FIG. 3 f is a chart of the shiftable, sheddable, and non-moveable demandfor a user cluster;

FIG. 3 g shows a demand pattern comparison (shiftable, sheddable, andnon-moveable demand in the same day) between two users in the same usercluster;

FIG. 3 h is a chart of exemplary DAM and RTM energy procurement costsfor a single day;

FIG. 3 i is a chart of the synthetic demand pattern (shiftable,sheddable, and non-moveable demand) used by a base case in a simulationcomparison with the method of FIG. 2;

FIG. 3 j is a chart of the DAM and RTM energy procurement costs used bya base case in a simulation comparison with the method of FIG. 2;

FIG. 3 k is a chart of the demand profile resulting from the use of themethod of FIG. 2 in a simulation comparison with a base case; and

FIG. 3 l is a chart of the procurement strategy resulting from the useof the method of FIG. 2 in a simulation comparison with a base case.

DETAILED DESCRIPTION

FIG. 1 is a block diagram of a typical smart power network 10(simplified) that operates to provide intelligent control and two-waycommunication among the various network 10 components. The network 10comprises a generating station(s) 12 that generateselectricity/electrical power from an energy source, such as a renewablesource (e.g., water, wind, etc.) or a non-renewable source (e.g., coal,natural gas, etc.). The generating station 12 may be configured in manyways and may utilize a combination of various energy sources. Thegenerating station 12 may also utilize energy that was stored for lateruse. The network 10 also comprises a transmission system 14 thatconverts the generated electrical power into high voltage power (via atransmission substation 14 a) and transfers the high voltage power overa long distance (via electrical transmission lines 14 b) to adistribution system 16. The distribution system 16 (via a powersubstation 16 a) converts the high voltage power into power at differentlower voltages and splits, or distributes, these lower-voltage powerflows off in multiple directions. The distribution system 16 may furthermodulate and regulate the lower-voltage power flows. Ultimately, thedistribution system 16 transfers the distributed power (via distributionlines 16 b) to electricity consumers or users 18. The users 18 aretypically categorized as residential users 18 a, commercial users 18 b,and industrial users 18 c. In many locations, the entities that generateelectricity are separate from the entities that locally distributeelectricity (such as a local utility company) so that the localdistributors (with their associated distribution systems 16) may also becategorized as a user though not a final or end user (or customer). Insome network 10 configurations, the distribution system 16 may alsotransfer the distributed power to sub-distribution systems (not shown)that further distribute the electricity to final or end users 18.

Each of the network 10 components will comprise and utilize appropriateequipment (for example, sensors, controls, computers, automation, smartmeters, etc.) that operates to implement the “smart” functionality ofthe network 10. The network 10 also comprises an operations center 20that permits intelligent management and control of the various network10 components and the overall network 10. The operations center 20comprises a two-way communications network 22 to connect to the variousnetwork 10 components for gathering and acting on information. Theoperations center 20 also comprises a computer system 24 to provide,among other functions, monitoring, reporting, supervising, andintelligence functions for managing and controlling the various network10 components and the overall network 10.

The various components of the network 10 are well understood smart gridcomponents. However, the computer system 24 of the operations center 20is also adapted to permit the network 10 to operate and to implementmethods in accordance with the present invention. FIG. 2 is a schematicrepresentation of a control method 100 implemented in accordance withthe present invention.

As will be explained in more detail below, the control method 100 is atwo step iterative process. In the first step, the load shedding problemis solved by assuming a known billing structure and a load shiftingprofile for each user/customer. This is done without knowing explicitlythe preferences of each user, but still reaching a solution that isoptimal for them. The second step uses this result to find the optimalbilling structure and load shifting profile. The method 100 iteratesbetween these two steps, reducing the cost of the solution at each stepand thus reaching optimality. Instead of solving a complex optimizationproblem, the method 100 breaks the problem into two simpler robustconvex optimization problems that can be even solved in a distributedfashion, thus reducing data transmission and processing requirements.

The framework for methods and systems of the present invention,including the above control method 100, is described as follows. Theobjective is to build a mechanism where a utility company has directcontrol of part of the energy consumption of their final customers giventhe energy they can procure from generators in the market. The presentinvention considers that the utility company has two tools to controlthe energy demand: demand shifting and demand shedding, and, further,that it procures the energy it delivers from a two-stage market with aday-ahead market (DAM) and a real-time market (RTM). Currently, thereare no methods or systems that involve two different mechanismssimultaneously to shape the demand, nor one that accounts for the energyprocurement in a two-level market.

In formulating a demand model, the present invention assumes that theutility company can monitor and control each individual customer, orcluster of customers, and that these customers can categorize theirenergy consumption in three different groups: loads that can be shed,those that can be shifted, and those that can neither be shed norshifted, i.e., the non-movable loads. Although it sounds reasonable thatthe consumption of each of the different client's appliances areassigned to a single load category, it is important to note that thereare many cases in which this is not the case. For example, take theenergy consumption of an air conditioning (A/C) unit. The energyconsumption level required for the A/C unit to give a minimum level ofcomfort could be categorized as a non-moveable load, whereas theadditional load required to get to the current temperature desired bythe customer can be categorized as a sheddable one. The presentinvention also assumes that the time window the utility company wants tooptimize can be divided into T equal-sized time steps, and, without lossof generality, it assumes T to be one day.

Continuing with the model, let q_(it), with i={1, . . . , n} and t={1, .. . , T} denote the total load of user i at time-step t; and thusq_(i)={q_(i1) . . . q_(iT)} denotes the total demand profile for user ifor that one day. At each time-step t, the demand of user i is composedby a sheddable demand q_(it) ^((d)), a shiftable demand q_(it) ^((s)),and a non-sheddable, non-shiftable demand q_(it) ^((n)), i.e.,q_(it=)q_(it) ^((d))+q_(it) ^((s))+q_(it) ^((n)).

As noted above, the sheddable demand is the part of the user's demandthat the utility company can shed, making the user incur a personal lossof value. For example, reducing the number of lights used to light anarea or increasing the target temperature of the A/C unit will drop thetotal consumption, bringing certain levels of discomfort to the user. Inall these cases, the demand is not shifted/postponed to a differenttime.

Thus, let y_(it) denote the amount of load user i sheds at time t, andy_(i) denote the shedding profile vector for user i. Given that attime-step t user i has a sheddable load of q_(it) ^((d)) (i.e., themaximum amount of sheddable load) then it follows that y_(it)≦q_(it)^((d)). In addition, let D_(i)(y_(it)) denote the discomfort cost foruser i for shedding a load y_(it), with D_(i):

₃₀→

₊. To incentivize this shedding, the utility company gives acompensation of p_(it) ^((d)) to user i at time-step t per unit of loadshed. Given the incentive p_(it) ^((d)) per unit of load, user i willdecide to shed an amount y_(it)=S_(i)(p_(it) ^((d))), with S_(i):

₊→

₊. In the article by L. Chen, N. Li, S. H. Low and J. C. Doylereferenced above, the authors use a similar model where they assumedthat the discomfort function D_(i)(y_(it)) is a continuous, increasing,strictly convex function, with D_(i)(0)=0, and that the sheddingfunction is of the form y_(it)=S_(i)(p_(it) ^((d)))=a_(i)p_(it) ^((d))where a_(i) is a user-dependent parameter.

As noted above, the shiftable demand is the part of the client's demandthat can be shifted during the day. Items like washing machines, dishwashers, and charging electric vehicles fall in this category sincethere is a window of time for using these machines and not a singlespecific point in the day. All these appliances can be used in a certaininterval within the T time-step window. Since the shiftable demandcannot be shed, the total shiftable demand for user i has a fixed valueΣ_(t)q_(it) ^((s))={circumflex over (q)}_(i) ^((s)). The shiftabledemand is composed by the consumption requirements of m differentappliances, where the consumption of appliance j of user i at time t isgiven by x_(ijt) and the time profile of the appliance is denoted byx_(ij.) Thus, at any given point in time the total consumption of user iis given by q_(it) ^((s))=Σ_(j)x_(ijt).

The total consumption of an appliance will be a given value denoted byΣ_(t)x_(ijt)={circumflex over (x)}_(ij), with a maximum value of x _(ij)and a minimum of x _(ij) at any given time-step t. For every appliancej, user i sets a time window [α_(ij), β_(ij)] in which the user irequires the appliance to run. For example, an electric vehicle couldcharge for 3 hours at any point between 10:00 p.m. and 7:00 a.m. Thisalso implies that x_(ijt)=0, ∀t∉└α_(ij), β_(ij)┘. In the article byA.-H. Mohsenian-Rad, et al. referenced above, the authors use a similarmodel for shifting user loads to achieve demand response. They also adda minimum stand-by requirement for x_(ijt), but since that can beconsidered as a non-moveable consumption the present framework adds thatto q_(ijt) ^((n)) and keeps only the shiftable portion in x_(ijt).

The non-moveable demand q_(it) ^((n)), as its name suggests, is the partof the demand that cannot be modified or has not been programmed ahead.The user requires that amount of energy at that specific time. Examplesof this are the demand of a refrigerator at the lowest viabletemperature setting or the on position in the A/C unit. Given that thisenergy consumption is not necessarily known ahead, q_(i) ^((n)) can beconsidered as a stochastic process. Further, it can assumed that theuser, or at least its smart meter, has some forecast {tilde over(q)}_(i) ^((n)) of the non-moveable demand for the whole time-window atthe beginning of the period. Using the ideas in the article by S. Meyn,M. Negrete-Pincetic, G. Wang, A. Kowli, and E. Hafieepoorfard referencedabove, the error between the meter's forecast and the real staticconsumption {circumflex over (q)}_(i) ^((n)), i.e., ε_(i) ^((n))={tildeover (q)}_(i) ^((n))−{circumflex over (q)}_(i) ^((n)), can be modeled asa driftless Brownian motion with variance σ_(i) ^((n)).

The demand model also addresses the billing system. At time-step t, useri is billed an amount B_(it) ^(S)(q_(it)) by the utility company, whereB_(it) ^(S)(q_(it)):

₊→

₊. In order to incentivize customers to set part of their demand asshiftable or sheddable demand, these loads are charged with a differentbilling structure B_(it) ^(f)(q_(it)), where B_(it) ^(f)(q_(it)):

₊→

₊. Since the utility company will be able to control these loadsdirectly, it is assumed that the billing structure does not depend on t,i.e., B_(it) ^(f)(q_(it))=B_(t) ^(f)(q_(it)), ∀t (∀ being a universalquantifier signifying “given any” or “for all”).

In formulating the procurement model, the utility company must be ableto satisfy the total demand Q_(t)=Σ_(i)q_(it), at every time-step t, andthus deliver a demand profile Q=[Q₁ . . . Q_(T)]. The utility companysatisfies this demand by procuring this energy from a two-stage marketcomposed by a day-ahead market (DAM) whose purchases are closed a day inadvance of the actual demand, and a real-time market (RTM) that clearsclose to real-time. The prices in these two markets can differsignificantly. Let C_(t) ^(d)(Q_(t)), with C_(t) ^(d)(Q_(t)):

² ₊→

₊ denote the total cost of procuring energy from the DAM and deliveringto satisfy a demand of Q_(t) at time t. Similarly, let C_(t)^(S)(Q_(t)), with C_(t) ^(S)(Q_(t)):

² ₊→

₊ denote the total cost of procuring energy from the RTM and deliveringto satisfy a demand of Q_(t) at time t. In general, we can assume thatC_(t) ^(d)(Q_(t))≦C_(t) ^(S)(Q_(t)) with C_(t) ^(d)(Q_(t)) being knownby the utility company and C_(t) ^(S)(Q_(t)) being an unknown stochasticprocess that represents the variation in prices that the RTM shows.

Within the framework described above, there are several differentproblems of interest for a utility company that can be addressed. Thefirst is the cost/revenue problem. Given the cost of procuring anddelivering load, C_(t) ^(d)(Q_(t)) and C_(t) ^(S)(Q_(t)), a billingstructure B_(it) ^(f)(q_(it)) and B_(it) ^(S)(q_(it)), ∀i, t andincentive prices per unit of load p_(i) ^((d)),∀i, what is the optimalload delivery that maximizes the utility company's revenue and satisfiesthe demand. This is similar to the problems posed in the articles byA.-H. Mohsenian-Rad et al. and L. Chen, N. Li, S. H. Low and J. C.Doyle, where the billing structure B_(it) ^(f)(q_(it)) is given and theauthors are interested in computing the demand pattern Q that maximizesthe revenues.

Another is the PAR problem that seeks to minimize the asset requirementsto deliver peak-energy, which can be achieved by minimizing thepeak-to-average ratio (PAR). This problem can be written as follows(equation 1):

$\begin{matrix}{{\min\limits_{X,Y}{\max\limits_{t}\frac{{TQ}_{t}}{\sum_{t}Q_{t}}}}{s.t.\text{:}}\begin{matrix}{{Q_{t} = {\sum_{i}q_{it}}},} & {{\forall t},} \\{{q_{it} = {\left( {q_{it}^{(d)} - y_{it}} \right) + q_{it}^{(s)} + {\overset{\_}{q}}_{it}^{(n)}}},} & {{\forall t},{\forall i}} \\{{q_{it}^{(s)} = {\sum_{j}x_{ijt}}},} & {{\forall t},{\forall i}} \\{{{\sum_{t}x_{ijt}} = {\hat{x}}_{ij}},} & {{\forall j},{\forall i}} \\{{x_{ijt} = 0},} & {{\forall j},{\forall i},{\forall{t \notin \left\lbrack {\alpha_{ij},\beta_{ij}} \right\rbrack}}} \\{{y_{it} \geq 0},} & {{\forall t},{\forall i}} \\{{y_{it} \leq q_{it}^{(d)}},} & {{\forall t},{\forall i}} \\{{x_{ijt} \geq {\underset{\_}{x}}_{ij}},} & {{\forall t},{\forall j},{\forall i}} \\{{x_{ijt} \leq {\overset{\_}{x}}_{ij}},} & {{\forall t},{\forall j},{\forall i}}\end{matrix}} & (1)\end{matrix}$

, where {tilde over (q)}_(i) ^((n)) the third line of equation 1)denotes the forecast the utility company has of the non-moveable demandfor the next day. This problem could also be posed as a robustoptimization problem, considering the error ε_(it) ^((n)) of theforecast for each user.

A more sophisticated problem is, given energy costs, billing structures,and incentive prices, to compute a solution that achieves a demandprofile Ψ=[Ψ₁ . . . Ψ_(T)] with a minimum cost. This is the demandshaping problem. For a given error parameter Δ, this problem can bewritten as follows (equation 2):

$\begin{matrix}{{{\min\limits_{X,Y}{\sum\limits_{t}{c_{t}\left( Q_{t} \right)}}} + {\sum\limits_{i}{\sum\limits_{t}{p^{(d)}y_{it}}}} - {\sum\limits_{i}{\sum\limits_{t}{B_{t}\left( q_{it} \right)}}} + {\Delta {{Q - \Psi}}_{p}}}{s.t.\text{:}}\begin{matrix}{{Q_{t} = {\sum_{i}q_{it}}},} & {{\forall t},} \\{{q_{it} = {\left( {q_{it}^{(d)} - y_{it}} \right) + q_{it}^{(s)} + {\overset{\_}{q}}_{it}^{(n)}}},} & {{\forall t},{\forall i}} \\{{q_{it}^{(s)} = {\sum_{j}x_{ijt}}},} & {{\forall t},{\forall i}} \\{{{\sum_{t}x_{ijt}} = {\hat{x}}_{ij}},} & {{\forall j},{\forall i}} \\{{x_{ijt} = 0},} & {{\forall j},{\forall i},{\forall{t \notin \left\lbrack {\alpha_{ij},\beta_{ij}} \right\rbrack}}} \\{{y_{it} \geq 0},} & {{\forall t},{\forall i}} \\{{y_{it} \leq q_{it}^{(d)}},} & {{\forall t},{\forall i}} \\{{x_{ijt} \geq {\underset{\_}{x}}_{ij}},} & {{\forall t},{\forall j},{\forall i}} \\{{x_{ijt} \leq {\overset{\_}{x}}_{ij}},} & {{\forall t},{\forall j},{\forall i}}\end{matrix}} & (2)\end{matrix}$

The present invention primarily focuses on the cost/revenue problem. Theobjective is to compute an optimal schedule for the shiftable demand andthe sheddable demand, such that the revenues are maximized given cost,billing, and price incentive structures. The present invention assumesseveral requirements for this problem. One requirement is fairness.Specifically, the billing structure must be fair. If two customersconsume the same amount at the same time, then the billing they shouldreceive shouldn't differ even though their usage utility functions mightdiffer. This means B_(i) ^(f)(q_(it))=B^(f)(q_(it)) and B_(it)^(S)(q_(it))=B_(t) ^(S)(q_(it)), ∀i. Similarly, if two users arerequested to shed the same amount of demand at the same time period t,then the payment they receive should be the same, that is p_(it)^((d))=p_(t) ^((d)), ∀i. Another requirement is proportionality. Thepresent invention assumes the more energy a customer consumes, thelarger the billing price should be, that is,B^(f)(q_(it1))≧B^(f)(q_(it2)) if q_(it1)≧q_(it2). Furthermore, thepresent invention assumes proportionality, that is,B^(f)(q_(it1))/B^(f)(q_(it2))=q_(it1)/q_(it2). This is certainlyjustified for small consumers/households, whereas it might not be truefor larger industrial consumers that pay additional fees for peakconsumption, load factors, and other factors. The same holds true forB_(t) ^(S)(q_(it)).

Another requirement is positive revenues. The present invention assumesthat the revenues for delivering the required demand Q are positive,that is, Σ_(t)B^(f)(Q_(t) ¹)+B_(t) ^(S)(Q_(t) ²)≧Σ_(t)C_(t) ^(d)(Q_(t)¹)+C_(t) ^(S)(Q_(t) ²). Another requirement relates to procurement. Let{tilde over (q)}_(i) ^((n))=[{tilde over (q)}_(i1) ^((n)) . . . {tildeover (q)}_(iT) ^((n))] denote an optimal procurement of the non-moveabledemand of user i in the DAM. The present invention assumes that energyprocurement is done in the following way. Since the utility companyknows the shiftable demand and the sheddable demand a priori (q_(i)^((S)) and q_(i) ^((d))) and an optimal procurement of the non-moveabledemand {tilde over (q)}_(i) ^((n)), it is assumed that it purchases atotal demand {tilde over (Q)}=[{tilde over (Q)}₁ . . . {tilde over(Q)}_(T)]{tilde over (Q)}_(t)=Σ_(t)(q_(it) ^((d))+q_(it) ^((s))+{tildeover (q)}_(it) ^((n)) in the DAM, and it has a cost of C_(t) ^(d)({tildeover (Q)}_(t)) to deliver it to the final users. The difference betweenthe purchased energy {tilde over (Q)}_(t) ^((n)) and the realnon-moveable demand Q_(t) ^((n)) is purchased in the RTM and it has atotal delivery cost of C_(t) ^(S)((Q_(t) ^((n))−{tilde over (Q)}_(t)^((n)))⁺). Note that this framework can also account for the spinningreserve (i.e., the generation capacity that is online but unloaded andthat can respond quickly to compensate for generation or transmissionoutages). This is done by purchasing in the DAM not only {tilde over(Q)}_(t), but (1+ε){tilde over (Q)}_(t) where ε denotes the additionalfraction purchased as spinning reserve. Then, the total cost ofprocuring the energy will be given by C_(t) ^(d)((1+ε){tilde over(Q)}_(t))+C_(t) ^(S)((Q_(t) ^((n))−(1+ε){tilde over (Q)}_(t)^((n))−εQ_(t) ^((d))−εQ_(t) ^((s)))⁺).

Another requirement relates to the shedding function. Most of thecurrent literature defines a simple shedding function S_(i)(p_(it)^((d))) since that simplifies the analysis significantly. For example,in the article by A.-H. Mohsenian-Rad, et al., the authors usey_(it)=S_(i)(p_(it) ^((d)))=a_(i)(p_(it) ^((d))), where a_(i) is auser-dependent parameter, and then proceed to compute an a_(i) thatmaximizes the user's revenues. There are two main issues with thisapproach. First, the user is changing its preference every time a priceis given and, second, it is already known that the user has a discomfortcost D_(i)(y_(it)) which can be used to find the optimal load y_(it)that will maximize the user's revenues. Hence, instead of using asimplification for the shedding function, the present invention solvesthe user's optimization problem to find the optimal load to shed.

With these requirements assumed, the present invention addresses thecost/revenue problem in the following way. For simplicity, letB_(t)(q_(it) ¹, q_(it) ²)=B^(f)(q_(it) ¹)+B_(t) ^(S)(q_(it) ²) andC_(t)(Q_(t) ¹, Q_(t) ²)=C_(t) ^(d)(Q_(t) ¹)+C_(t) ^(S)(Q_(t) ²).Similarly, let X=[x′₁₁ . . . x′_(nm)]′ and Y=[y′₁ . . . y′_(n)]′. Withthis, the optimization problem for the utility company becomes asfollows (equation 3):

$\begin{matrix}{{{\max\limits_{X,Y,{\overset{\sim}{Q}}_{t}^{(n)}}{\sum\limits_{t}{\sum\limits_{i}\left\lbrack {{B_{t}\left( {{q_{it}^{(s)} + q_{it}^{(d)} - y_{it}},q_{it}^{(n)}} \right)} - {p_{t}^{(d)}y_{it}}} \right\rbrack}}} - {C_{t}\left( {{Q_{t}^{(d)} - Y_{t} + Q_{t}^{(s)} + {\overset{\sim}{Q}}_{t}^{(n)}},{Q_{t}^{(n)} - {\overset{\sim}{Q}}_{t}^{(n)}}} \right)}}\mspace{20mu} {s.t.\text{:}}\mspace{20mu} \begin{matrix}{{q_{it}^{(s)} = {\sum_{j}x_{ijt}}},} & {{\forall t},{\forall i}} \\{{{\sum_{t}x_{ijt}} = {\hat{x}}_{ij}},} & {{\forall j},{\forall i}} \\{{x_{ijt} = 0},} & {{\forall j},{\forall i},{\forall{t \notin \left\lbrack {\alpha_{ij},\beta_{ij}} \right\rbrack}}} \\{{y_{it} \geq 0},} & {{\forall t},{\forall i}} \\{{y_{it} \leq q_{it}^{(d)}},} & {{\forall t},{\forall i}} \\{{x_{ijt} \geq {\underset{\_}{x}}_{ij}},} & {{\forall t},{\forall j},{\forall i}} \\{{x_{ijt} \leq {\overset{\_}{x}}_{ij}},} & {{\forall t},{\forall j},{\forall i}}\end{matrix}} & (3)\end{matrix}$

This optimization problem assumes that the actual procurement cost inthe RTM C_(t) ^(S)(Q_(t)) and the actual non-moveable demand Q_(t)^((n)) are known a priori. Since that is seldom the case, a robustoptimization approach is used to compute an optimal solution. The ideabehind a robust optimization formulation is to consider not a singleexpected value of a stochastic variable, but a whole set of possiblevalues from which the approach needs to cover the solution. The amountof robustness in the solution is then controlled by the size of theuncertainty set in which the stochastic variables are considered to beincluded.

Given that there are two main stochastic variables in the aboveoptimization problem (equation 3), then the optimization problem canhave robustness in two different ways. First, there can be robustness inprocurement costs. The RTM prices can be considered as a stochasticprocess, hence, the problem becomes how much energy to purchase in theDAM given that there is uncertainty in the prices in the RTM. In thiscase, the uncertainty set (symbolized as calligraphic C)

C_(t), for t={1, . . . , T}

will represent the possible cost functions the utility company will getin the RTM at time step t during the next day. Second, there can berobustness in consumption level. The second robust optimizationformulation is achieved by considering the stochastic process behind thenon-moveable demand. In this case

Q _(t) ^((n)), for t={1, . . . , T}

denotes the uncertainty set (symbolized as calligraphic Q) for thenon-moveable demand Q_(t) ^((n)). The constraint that demand is within acertain convex set is added and an optimal solution for all demandswithin that set is computed.

Both settings can be combined and optimal solutions can be computed whenthe procurement cost and the consumption are unknown. This might lead tovery conservative solutions since the formulation accounts for twosources of uncertainty. The full robust problem formulation is thengiven by (equation 4):

$\begin{matrix}{{{\max\limits_{X,Y,{\overset{\sim}{Q}}_{t}^{(n)}}{\min\limits_{C_{t},Q_{t}^{(n)}}{\sum\limits_{t}{\sum\limits_{i}\left\lbrack {{B_{t}\left( {{q_{it}^{(s)} + q_{it}^{(d)} - y_{it}},q_{it}^{(n)}} \right)} - {p_{t}^{(d)}y_{it}}} \right\rbrack}}}} - {C_{t}\left( {{Q_{t}^{(d)} - Y_{t} + Q_{t}^{(s)} + {\overset{\sim}{Q}}_{t}^{(n)}},{Q_{t}^{(n)} - {\overset{\sim}{Q}}_{t}^{(n)}}} \right)}}\mspace{20mu} {s.t.\text{:}}\mspace{20mu} \begin{matrix}{{q_{it}^{(s)} = {\sum_{j}x_{ijt}}},} & {{\forall t},{\forall i}} \\{{{\sum_{t}x_{ijt}} = {\hat{x}}_{ij}},} & {{\forall j},{\forall i}} \\{{x_{ijt} = 0},} & {{\forall j},{\forall i},{\forall{t \notin \left\lbrack {\alpha_{ij},\beta_{ij}} \right\rbrack}}} \\{{C_{t} \in C_{t}},} & {{\forall t},} \\{{Q_{t}^{(n)} \in Q_{t}^{(n)}},} & {{\forall t},} \\{{y_{it} \geq 0},} & {{\forall t},{\forall i}} \\{{y_{it} \leq q_{it}^{(d)}},} & {{\forall t},{\forall i}} \\{{x_{ijt} \geq {\underset{\_}{x}}_{ij}},} & {{\forall t},{\forall j},{\forall i}} \\{{x_{ijt} \leq {\overset{\_}{x}}_{ij}},} & {{\forall t},{\forall j},{\forall i}}\end{matrix}} & (4)\end{matrix}$

The level of robustness in the above problem (equation 4), and thus howconservative is the solution, will depend on the uncertainty sets

C_(t) and Q_(t) ^((n)).

In general, these uncertainty sets can be either boxes, which basicallyadd a set of box constraints to the optimization model, or ellipsoids.The main advantage of box constraints is its ease of implementationsince most of the time that is translated to interval constraints foreach variable. The drawback is that the corners of the box make thesolution overly conservative since they are much further away from thecentre of the box compared to the sides. On the other hand, ellipsoidsets give better solutions and don't suffer from beingoverly-conservative. The disadvantage is that they are harder toimplement since they are not linear and can only be approximated if alinear model is desired.

Considering the cost/revenue problem detailed above, there is also aneed to design a mechanism that would determine the billing structureB_(t)(q_(it) ¹, q_(it) ²) and the price incentive structure p_(t)^((d))), such that the whole system achieves equilibria. That is, giventhe structure computed by this mechanism, the optimal solution obtainedby the utility company will give no incentives to the final customers tochange their solutions or cheat.

As a separate issue, to understand what will be the objective of thefinal customer, the optimization problem they see must be posed. Attime-step t, user i is billed an amount B_(t)(q_(it) ^((s))+q_(it)^((d))+q_(it) ^((n))) by the utility company, and is paid p_(it) ^((d))per unit of load shed, hence the user will solve the followingoptimization problem (equation 5),

$\begin{matrix}{{{\min\limits_{x_{i},y_{i}}{\sum\limits_{t = 1}^{T}{B_{t}\left( {{q_{it}^{(s)} + q_{it}^{(d)} - y_{it}},q_{it}^{(n)}} \right)}}} + {D_{it}\left( y_{it} \right)} - {p_{it}^{(d)}y_{it}}}{s.t.\text{:}}\begin{matrix}{{q_{it}^{(s)} = {\sum_{j}x_{ijt}}},} & {{\forall t},} \\{{{\sum_{t}x_{ijt}} = {\hat{x}}_{ij}},} & {{\forall j},} \\{{x_{ijt} = 0},} & {{\forall j},{\forall{t \notin \left\lbrack {\alpha_{ij},\beta_{ij}} \right\rbrack}}} \\{{y_{it} \geq 0},} & {{\forall t},} \\{{y_{it} \leq q_{it}^{(d)}},} & {{\forall t},} \\{{x_{ijt} \geq {\underset{\_}{x}}_{ij}},} & {{\forall t},{\forall j},} \\{{x_{ijt} \leq {\overset{\_}{x}}_{ij}},} & {{\forall t},{\forall j},}\end{matrix}} & (5)\end{matrix}$

In summary, the cost/revenue problem has two basic parts that need to beaddressed. One part is that prices and billing structures that achievesome equilibrium must be computed, and, in a second part, the allocationof demand for each user given these prices and billing structure must becomputed. The control method 100 solves this problem by performing atwo-step process which iterates between computing a billing structureand computing the price incentive to achieve an equilibrium.

To analyze how to solve the general version of this problem posed inequation 4, additional structure is pointed out. First, it is assumedthat the general billing structure is linear, that is,B^(f)(q_(it))=b^(f)q_(it) and B_(t) ^(S)(q_(it))=b_(t) ^(S)q_(it), forall t, where b^(f) and b_(t) ^(S) and are constants. Although this isnot the case for large industrial consumers, most residential householdsdo have a linear billing cost. As noted above, since the utility companyhas direct control over the shiftable demand and the sheddable demand,it is assumed that the billing structure for these demands is constantthroughout the day, i.e., b_(t) ^(f)=b^(f), for all t.

The method 100 comprises a first step 102 that addresses price incentiveand shedding calculation. The objective of this first step 102 is tocompute the optimal amount of demand to shed and the price incentiverequired to do so. It is assumed that the billing structure B_(t)(Q_(t)¹, Q_(t) ²) is known for all t, as well as the solution X for theshiftable loads. With these assumptions, and given p^((d)), the utilitycompany's problem (given by equation 4) simplifies to the following(equation 6):

$\begin{matrix}{{{\max\limits_{Y,{\overset{\sim}{Q}}_{t}^{(n)}}{\min\limits_{C_{t},Q_{t}^{(n)}}{\sum\limits_{t}{\sum\limits_{i}\left\lbrack {{b^{f}\left( {q_{it}^{(d)} - y_{it}} \right)} + {b_{t}^{s}q_{it}^{(n)}} - {p_{t}^{(d)}y_{it}}} \right\rbrack}}}} - {C_{t}\left( {{Q_{t}^{(d)} - Y_{t} + Q_{t}^{(s)} + {\overset{\sim}{Q}}_{t}^{(n)}},{Q_{t}^{(n)} - {\overset{\sim}{Q}}_{t}^{(n)}}} \right)}}\mspace{20mu} {s.t.\text{:}}\mspace{20mu} \begin{matrix}{{q_{it}^{(s)} = {\sum_{j}x_{ijt}}},} & {{\forall t},{\forall i}} \\{{C_{t} \in C_{t}},} & {{\forall t},} \\{{Q_{t}^{(n)} \in Q_{t}^{(n)}},} & {{\forall t},} \\{{y_{it} \geq 0},} & {{\forall t},{\forall i}} \\{{y_{it} \leq q_{it}^{(d)}},} & {{\forall t},{\forall i}}\end{matrix}} & (6)\end{matrix}$

, which is separable in t; hence, for every time step t the followingoptimization problem needs to be solved (step 104) (equation 7):

$\begin{matrix}{{{\max\limits_{Y_{t},{\overset{\sim}{Q}}_{t}^{(n)}}{\min\limits_{C_{t},Q_{t}^{(n)}}{\sum\limits_{i}\left\lbrack {{b^{f}\left( {q_{it}^{(d)} - y_{it}} \right)} + {b_{t}^{s}q_{it}^{(n)}} - {p_{t}^{(d)}y_{it}}} \right\rbrack}}} - {C_{t}^{d}\left( {Q_{t}^{(d)} - Y_{t} + Q_{t}^{(s)} + {\overset{\sim}{Q}}_{t}^{(n)}} \right)} - {C_{t}^{s}\left( {Q_{t}^{(n)} - {\overset{\sim}{Q}}_{t}^{(n)}} \right)}}\mspace{20mu} {s.t.\text{:}}\mspace{20mu} \begin{matrix}{{q_{it}^{(s)} = {\sum_{j}x_{ijt}}},} & {\forall i} \\{{C_{t} \in C_{t}},} & \; \\{{Q_{t}^{(n)} \in Q_{t}^{(n)}},} & \; \\{{y_{it} \geq 0},} & {\forall i} \\{{y_{it} \leq q_{it}^{(d)}},} & {\forall i}\end{matrix}} & (7)\end{matrix}$

In the setting that the price incentive is set externally (e.g., bygovernment regulations, contracts, etc.), and the utility company canshed the load directly, this optimization problem can be solved usingstandard convex optimization techniques (step 104 a), and the method 100can move to a second step 110 for billing and shift calculation.

In the setting that the utility company has to set the price so that thefinal users are motivated to shed their load, the utility company has nodirect control over the load, and can only set the price offer so thatrespective pre-programmed smart meters at a user take the decision ofwhat amount of load to shed depending on the user's preferences and lossfunction (step 104 b). From the user's point of view, given a billingstructure, a solution for the shiftable load, and a price incentivep_(t) ^((d)), the optimization problem for every time step t is given by(equation 8):

$\begin{matrix}{{{\min\limits_{y_{i}}\; {b^{f}\left( {- y_{it}} \right)}} + {D_{it}\left( y_{it} \right)} - {p_{t}^{(d)}y_{it}}}{s.t.\text{:}}{{y_{it} \leq q_{it}^{(d)}},{y_{it} \geq 0.}}} & (8)\end{matrix}$

, which is easily solvable once D_(it) is known. If it is assumed D_(it)is continuous, increasing, and differentiable, then the solution forthis user optimization problem (equation 8) is given by (equation 9):

y* _(it) =S _(it)(p_(t) ^((d)))=min{q _(it) ^((d)),max{0,D′ _(it) ⁻¹(p_(t) ^((d)) +b ^(f))}}  (9)

This computes the optimal value of y_(it) for each user i at each timestep t, and thus can be considered as the shedding function for user iat time t. Let S_(t)(p_(t) ^((d)))=Σ_(i)S_(it)(p_(t) ^((d))), then theutility company's problem for each time step t is given by (step 104 b)(equation 10):

$\begin{matrix}{{{\max\limits_{p_{t}^{(d)},{\overset{\sim}{Q}}_{t}^{(n)}}{\min\limits_{C_{t},Q_{t}^{(n)}}{\sum\limits_{i}\left\lbrack {{b^{f}q_{it}^{(d)}} + {b_{t}^{s}q_{it}^{(n)}}} \right\rbrack}}} - {\left( {p_{t}^{(d)} + b^{f}} \right)Y_{t}} - {C_{t}\left( {{Q_{t}^{(d)} - Y_{t} + Q_{t}^{(s)} + {\overset{\sim}{Q}}_{t}^{(n)}},{Q_{t}^{(n)} - {\overset{\sim}{Q}}_{t}^{(n)}}} \right)}}\mspace{20mu} {s.t.\text{:}}\mspace{20mu} \begin{matrix}{{q_{it}^{(s)} = {\sum_{j}x_{ijt}}},} & {\forall i} \\{{Y_{t} = {S_{t}\left( p_{t}^{(d)} \right)}},} & \; \\{{C_{t} \in C_{t}},} & \; \\{{Q_{t}^{(n)} \in Q_{t}^{(n)}},} & \; \\{{y_{it} \geq 0},} & {\forall i} \\{{y_{it} \leq q_{it}^{(d)}},} & {\forall{i.}}\end{matrix}} & (10)\end{matrix}$

If S_(t)(p_(t) ^((d))) is known then again this can be directly solved,although there is no assurance that the problem is convex any more sinceS_(t)(p_(t) ^((d))) might not be convex. Knowing S_(t)(p_(t) ^((d))) isa limiting assumption since it implies that either all the user lossfunctions D_(it)(y_(it)) are known (which has important privacyconcerns) or that S_(t)(p_(t) ^((d))) can be recovered by sending theprice information to the users' smart meters (which might be a dataintensive operation but doesn't have to be done often).

The second step 110 for billing and shift calculation uses the resultsfrom the previous step 102, i.e., p^((d)) and y_(i), ∀i and computes thebilling structure B_(t), and the demand schedule for the shiftable loadx_(ij), ∀i, j. Since p^((d)) and y_(i), ∀i, are known, the utilitycompany's problem is simplified to the following (step 112):

$\begin{matrix}{{{\max\limits_{X,Y,{\overset{\sim}{Q}}_{t}^{(n)}}{\min\limits_{C_{t},Q_{i}^{(n)}}{\sum\limits_{t}{\sum\limits_{i}{B_{t}\left( {{q_{it}^{(s)} + q_{it}^{(d)} - y_{it}},q_{it}^{(n)}} \right)}}}}} - {C_{t}\left( {{Q_{t}^{(d)} - Y_{i} + Q_{t}^{(s)} + {\overset{\sim}{Q}}_{i}^{(n)}},{Q_{t}^{(n)} - {\overset{\sim}{Q}}_{t}^{(n)}}} \right)}}\mspace{20mu} {s.t.\text{:}}\mspace{20mu} \begin{matrix}{{q_{it}^{(s)} = {\sum_{j}x_{ijt}}},} & {{\forall t},{\forall i}} \\{{{\sum_{t}x_{ijt}} = {\hat{x}}_{ij}},} & {{\forall j},{\forall i}} \\{{x_{ijt} = 0},} & {{\forall j},{\forall i},{\forall{t \notin \left\lbrack {\alpha_{ij},\beta_{ij}} \right\rbrack}}} \\{{C_{t} \in C_{t}},} & {{\forall t},} \\{{Q_{t}^{(n)} \in Q_{t}^{(n)}},} & {{\forall t},} \\{{x_{ijt} \geq {\underset{\_}{x}}_{ij}},} & {{\forall t},{\forall j},{\forall i}} \\{{x_{ijt} \leq {\overset{\_}{x}}_{ij}},} & {{\forall t},{\forall j},{\forall i}}\end{matrix}} & (11)\end{matrix}$

There are two main approaches to solve the problems as the one posed inequation 11. One is to assume a utility function for each user that willrelate the energy consumption with the price of consuming that energy,and thus control the billing. The problem with this approach in practiceis that it is very hard to recover or even estimate these utilityfunctions. The other relies on the fact that since the electric marketis highly regulated, in many cases the billing costs to final users areregulated as well and to be kept within certain values. The method 100utilizes the second approach and assumes that the utility ratio Υ(symbolized as the Greek letter upsilon with hooks and as seen inEquations 12, 14 and 15 below) is fixed. The latter is similar to thearticle by A.-H. Mohsenian-Rad, et al., but unlike this previous work,the method 100 keeps the ratio separately for the shiftable plussheddable load and the non-moveable load. Hence, the formulation is asfollows:

$\begin{matrix}{{{\mathrm{\Upsilon} \equiv \frac{\sum_{t}{\sum_{i}{B^{f}\left( {q_{it}^{(s)} + q_{it}^{(d)} - y_{it}} \right)}}}{\Psi {\sum_{t}{C_{t}^{d}\left( {Q_{t}^{(d)} - Y_{t} + Q_{t}^{(s)} + {\overset{\sim}{Q}}_{t}^{(n)}} \right)}}}} = {\frac{\sum_{i}{B_{t}^{s}\left( q_{it}^{(n)} \right)}}{{\Phi_{i}{C_{t}^{d}\left( {Q_{t}^{(d)} - Y_{i} + Q_{t}^{(s)} + {\overset{\sim}{Q}}_{i}^{(n)}} \right)}} + {C_{i}^{s}\left( {Q_{i}^{(n)} - {\overset{\sim}{Q}}_{t}^{(n)}} \right)}} \geq 1}},} & (12)\end{matrix}$

where the second equality is for all t and

$\begin{matrix}{{{\Psi = \frac{{\sum_{t}Q_{i}^{(d)}} - Y_{t} + Q_{i}^{(s)}}{{\sum_{t}Q_{t}^{(d)}} - Y_{i} + Q_{t}^{(s)} + Q_{i}^{(n)}}};}{{\Phi_{i} = \frac{{\overset{\sim}{Q}}_{i}^{(n)}}{{\sum_{i}Q_{t}^{(d)}} - Y_{t} + Q_{t}^{(s)} + {\overset{\sim}{Q}}_{t}^{(n)}}},{\forall{t.}}}} & (13)\end{matrix}$

Using equations 12 and 13, respective billing structures may beformulated as follows (step 114 a):

$\begin{matrix}{\mspace{79mu} {{b^{f} = \frac{\mathrm{\Upsilon}{\sum_{t}{C_{t}^{d}\left( {Q_{i}^{(d)} - Y_{i} + Q_{t}^{(s)} + {\overset{\sim}{Q}}_{t}^{(n)}} \right)}}}{{\sum_{t}Q_{i}^{(d)}} - Y_{t} + Q_{t}^{(s)} + {\overset{\sim}{Q}}_{i}^{(n)}}},\mspace{20mu} {and}}} & (14) \\{{b_{t}^{s} = {{\left( \frac{\mathrm{\Upsilon}\; {\overset{\sim}{Q}}_{t}^{(n)}}{Q_{i}^{(n)}} \right)\frac{C_{t}^{d}\left( {Q_{i}^{(d)} - Y_{t} + Q_{i}^{(s)} + {\overset{\sim}{Q}}_{t}^{(n)}} \right)}{{\sum_{t}Q_{i}^{(d)}} - Y_{t} + Q_{i}^{(s)} + Q_{t}^{(n)}}} + {\left( \frac{\mathrm{\Upsilon}}{Q_{i}^{(n)}} \right){C_{i}^{s}\left( {Q_{i}^{(n)} - {\overset{\sim}{Q}}_{t}^{(n)}} \right)}}}},{\forall{t.}}} & (15)\end{matrix}$

Note that the value of b^(f) can be computed without knowing the optimalsolution for X, since only Σ_(t) Σ_(i) Σ_(j)x_(ijt) is needed, which isknown and constant. On the other hand, b_(t) ^(S) cannot be computeduntil the realization of the non-moveable demand; but, since X is notaffected by b_(t) ^(S) one can compute the optimal shiftable demand bysolving the following (step 114 b):

$\begin{matrix}{{{\max\limits_{X,Q_{t}^{(n)}}{\min\limits_{C_{i},Q_{t}^{(n)}}{\sum\limits_{t}{\sum\limits_{i}{b^{f}q_{it}^{(s)}}}}}} - {\sum\limits_{t}{C_{t}^{d}\left( {Q_{i}^{(d)} - Y_{i} + Q_{t}^{(s)} + {\overset{\sim}{Q}}_{t}^{(n)}} \right)}} - {\sum\limits_{t}{C_{t}^{s}\left( {Q_{t}^{(n)} - Q_{t}^{(n)}} \right)}}}\mspace{20mu} {s.t.\text{:}}\mspace{20mu} \begin{matrix}{{q_{it}^{(s)} = {\sum_{j}x_{ijt}}},} & {{\forall t},{\forall i}} \\{{{\sum_{t}x_{ijt}} = {\hat{x}}_{ij}},} & {{\forall j},{\forall i}} \\{{x_{ijt} = 0},} & {{\forall j},{\forall i},{\forall{t \notin \left\lbrack {\alpha_{ij},\beta_{ij}} \right\rbrack}}} \\{{C_{t} \in C_{t}},} & {{\forall t},} \\{{Q_{t}^{(n)} \in Q_{t}^{(n)}},} & {{\forall t},} \\{{x_{ijt} \geq {\underset{\_}{x}}_{ij}},} & {{\forall t},{\forall j},{\forall i}} \\{{x_{ijt} \leq {\overset{\_}{x}}_{ij}},} & {{\forall t},{\forall j},{\forall i}}\end{matrix}} & (16)\end{matrix}$

With this problem solved, the method 100 returns to the price incentiveand shedding step 102 and repeats all of the method 100 steps untilconvergence is achieved (step 130). Importantly, the method 100 solvesthis problem in a centralized manner (unlike the distributed mannerpresented by the article by A.-H. Mohsenian-Rad, et al.).

From a procurement standpoint, the method 100 considers a two levelmarket from which the utility company purchased the energy required tocover the demand. Second, unlike previous work, the method 100 takesinto account that demand level and the procurement prices are indeedunknown stochastic processes. The method 100 accommodates these factorsthrough robust optimization. Finally, the method 100 simultaneously usesdemand shedding and demand shifting as a means to shape the demandaccording to the required objectives. Although three different problemswere identified that can be tackled in the framework of the presentinvention, the method 100 focuses on the revenue optimization problemwhich is the most important one for utility companies.

In simulations using randomly-generated data, the results of using thecontrol method 100 have been compared against the results of using nocontrol method. The test procedure is described below.

In the simulations, synthetic data was randomly generated for eachuser/customer. To get useful and more realistic demand patterns,household consumption data was used as a seed to generate the syntheticdata for the users/customers. Eighteen (18) months of hourly data forsix (6) different users was used to compute seeds that would latergenerate random instances. In order to define the types of seeds, thedata was statistically analyzed in several different ways. FIGS. 3 a and3 b show two different examples. In FIG. 3 a, the hourly consumptiondata was grouped for a single user for every month and its distributionwas computed, and are presented as vertical histograms, one for eachmonth. Similarly, FIG. 3 b shows the result of grouping the data of asingle user for a single day of the week (Tuesday in this case) for eachhour of the day, and are presented as vertical histograms. It is notedthere is a notable change in the consumption pattern depending on thehour of the day, which also varies depending on the day of the week.

By grouping demand by day of the week and hour, a good balance wasobtained between recovering the details of the demand distribution andhaving enough data points to have a meaningful empirical distributionsince adding month, for example, would leave only 4 or 5 data points foreach group which cannot give enough information.

As FIG. 3 c shows, it is also noted that there are differences among thedifferent users in the raw data, although it is difficult to be certaingiven the low amount of samples. Still, it was assumed that each userrepresents the average or typical distribution of a certain cluster ofusers, and this was used as a seed to generate a population of usersthat has a mix between the different clusters.

Using this information as a seed, the hourly demand pattern for anysingle day of the week can be generated for a population that contains amix of the six (6) different users identified in the raw data. This canbe done by generating uniformly distributed random variables in thefollowing manner. Let f_(u),_(d,t)(z) denote the empirical probabilitymass function of the demand z of users in cluster

u ε {1, . . . , 6},

on day of the week d={1, . . . , 7}, and time

t ε {0, . . . , 23},

that were computed from the raw data, and let F_(u),_(d,t)(z) denote itscumulative distribution function. Then, given a day of the week D foreach user/customer of cluster U, twenty-four (24) uniformly distributedrandom variables z₀, . . . , z₂₃ are generated and the demand of thatuser i at time t is set equal to q_(it)=F_(U,D,t) ⁻¹(z_(t)), makingq_(it) a random variable with probability mass function f_(u),_(d,t)(z).

Although this method is simple to implement and allows the generation ofover a million users very quickly, it has the drawback that the demandof the users will be independent, which is not the case in real lifesince there is a positive correlation of the energy demand among users.The process to generate correlated random variables is more complicatedand requires the definition of a variance-covariance matrix for theusers, limiting the number of final users that can be simulated due tothe physical memory requirements in the test procedure. The currentimplementation can generate up to 5,000 users/customers, being limitedmainly by the simulation computer's RAM. In this case, when generatingan instance, a symmetric, positive definite matrix is randomly createdthat has only positive correlation factors among variables, S for eachcluster within the population that is being simulated.

It is important to note that the final generated random variables (thatis, q_(it)) will not necessarily have S as its variance-covariancematrix since F_(u),_(d,t)(z) could not preserve the correlation, but itserves as a simple and good enough proxy to positively correlate thesimulated demand patterns for the users of a cluster. Also, it is notedthat due to the correlation, the distribution of the demand of all usersof a cluster u in a certain day of the week d and hour t will not followthe corresponding distribution f_(u),_(d,t). But, as shown in FIG. 3 d,if several samples of the same user for the same day of the week andhour are taken, the final distribution for that user (bottom chart) willbe identical to the empirical one (top chart), just as expected.

FIG. 3 e shows the simulated average user demand for each cluster andfor a whole day. Finally, for each user, there needs to be a simulationof what part of that demand is shiftable, sheddable, or non-moveable atevery time in the day simulation. Since those patterns might differ fromone user to another, only the relative weights of each type of demandwere set for each cluster of users and the proportion for each finaluser, using the relative weights as a seed, were generated.

FIG. 3 f shows the total aggregated demand for cluster 4 indicatingwhich part is considered shiftable, sheddable, or non-moveable. To showthat each user within this cluster actually has its own randomlygenerated demand, FIG. 3 g shows the daily demand pattern of twodifferent users within the users of cluster 4 showing that although theyare different they do have some correlation; whereas, the proportions ofshiftable, sheddable, and non-moveable demands are completely randomamong users.

In the simulations, another important variable to test the method 100 isthe energy procurement cost. The simulations used data from the ElectricReliability Council of Texas website (http://www.ercot.com),specifically the Houston prices, to simulate the DAM and RTM prices. Twomonths of price data were used and the average for the corresponding dayof the week in which the simulation is done was used. The DAM prices aregiven hourly, but given that the RTM prices are provided with 15 minutesresolution, the maximum price for that hour was used as the hourly datato capture the worst case scenarios. FIG. 3 h shows an example of theDAM and RTM prices for a Friday.

In the simulations, the method 100 output is compared with a base casein order to analyze how good is the solution given by the method 100. Asa base case, the simulations consider the setting in which the utilitycompany has no control over the demand of its users and will justpurchase energy in the DAM using a forecast of the total energyconsumption for the next day and then buy in the RTM whatever extraenergy it requires over what was forecasted. Since the performance willdepend on how good the forecast is, for a first set of simulations, itwas assumed the non-moveable demand and the RTM prices were known at themoment the problem was solved. In other words, the robustness part waseliminated by setting the RTM prices uncertainty set and the Q_(t)^((n)) uncertainty sets equal to the actual values of those quantities.For this case, FIG. 3 i shows the total value of one of the randomlygenerated demands. Although in the base case the utility company cannotcontrol the final user's demand, the different types of demand(shiftable, sheddable, and non-moveable) are highlighted to compare itwith the method 100 output.

Given this demand profile, the base case will procure all its energy onthe DAM as shown in FIG. 3 j. Since no forecasts are involved there isno over-purchase in the DAM.

Finally, the utility ratio Υ=1.2 is set for all the simulations to keepthe same income ratio in the utility company. Then, the costs andbilling rates for the base case are as follows:

Output of Base Case

-   Total Procurement Cost: $ 603087.81-   Par: 2.6089-   Flat Rate to Consumers: $ 38.54/unit

The performance of the method 100 in simulation is now described. Themethod 100 is started with b^(f) and b_(t) ^(S) equal to the billingrate computed in the base case and the initial value for X_(t) equal tothe shiftable demand Q_(t) ^((s)). FIG. 3 k shows the result of five (5)iterations of the method 100. When compared to FIG. 3 i, it is clearthat the method 100 moves the shiftable demand to the hours where theenergy is cheaper and, when convenient, it also sheds some demand toreduce the energy procurement. FIG. 3 l shows how the energy procurementis made. Since the method 100 can decide when to purchase part of thenon-moveable demand, the procurement occurs in the RTM when the pricesare lower than the DAM.

Finally, the following is the output of the method 100 in which thereare several important things to note. First, the method 100 convergesfairly quickly, in just five (5) iterations. Second, the equilibriumresults in a solution that is more than 20% cheaper than the base casein terms of procurement costs while at the same time it reduces thebilling rates to consumers for the sheddable and shiftable loads, aswell as for most of the non-moveable demand, (except when theprocurement prices are higher). This last benefit is achieved thanks tothe important reduction in procurement costs. Finally, it is noted thePAR is also slightly reduced for this simulation. ioThis might not bethe case every time, and it depends on how steep are the changes inprices and the amount of available shiftable demand. In any case, thisat least shows that if the PAR is included in the objective it couldresult in a much better solution in terms of PAR.

Output of Method 100

-   Step 1—Total Cost: $ 471647.63-   Step 2—Total Cost: $ 477478.68-   Step 3—Total Cost: $ 477454.26-   Step 4—Total Cost: $ 477454.09-   Step 5—Total Cost: $ 477454.08-   Total Procurement Cost: $ 477454.08-   Improvement: 20.8317%-   New Par: 2.2083-   Flat Rate to Consumers (bf): $ 32.47/unit-   Spot Rate to Consumers (bs):-   24.8385-   31.6698-   31.6698-   31.6698-   31.6698-   31.6698-   31.6698-   31.6698-   31.6698-   31.6698-   31.6698-   31.6698-   31.6698-   31.6698-   46.0320-   31.6698-   84.2850-   43.1370-   39.2160-   36.6045-   35.0310-   31.6698-   31.6698-   31.6698

The present invention provides a modeling framework and a control method100 (expressed in part as an optimization algorithm) for demand shapingthrough load shedding and shifting in a electrical smart grid. Thepresent invention tackles the problem of controlling the flow of energy,for example, through demand shaping, in a much more general, and novel,framework than the current state of the art. First, unlike currentmechanisms, the present invention considers two tools simultaneously forshaping the demand: shedding and shifting of load. Through direct loadcontrol the utility company will schedule/shift part of the demand andthrough price incentives it controls the level of load shedding.Additionally, the present invention considers the two-level market inwhich utility companies purchase their energy, the day-ahead market andthe real-time market, which have different cost structures, and thusaddresses the problem of how much energy to purchase on each to coverthe demand. Finally, the different optimization sub-problems that mustbe solved are all posed in a robust optimization framework, allowing thepresent invention to accommodate the fact that both the final demand andthe costs in the real-time market are unknown stochastic processes andnot the deterministic values assumed by most of the literature. Thefinal objective of the present invention is then to find the bestshedding and shifting profile and how much energy to purchase in eachmarket to satisfy the demand while minimizing costs.

Other modifications are possible within the scope of the invention. Forexample, the smart power network 10 may be a power sub-network connectedto a larger utility power network or even a private power network. Also,the power network 10 and its components have been described in asimplified fashion and may each be constructed in various well-knownmanners and using various well-known components. Also, the controlmethod 100 may be utilized by energy generators as well as energydistributers since energy generators also buy energy from the energymarkets but they also generate it themselves with completely differentcost structures.

Also, although the steps of the control method 100 have been describedin a specific sequence, the order of the steps may be re-ordered in partor in whole and the steps may be modified, supplemented, or omitted asappropriate. Also, the power network 10 and the computer system 24 mayuse various well known algorithms and software applications to implementthe steps and substeps. Further, the control method 100 may beimplemented in a variety of algorithms and software applications.

What is claimed is:
 1. A method for controlling the flow of energy in anelectrical power grid, comprising: a. obtaining an optimal load sheddingprofile for a user of the grid; b. obtaining an optimal billingstructure and an optimal load shifting profile for the user; c.iterating the steps of obtaining an optimal load shedding profile andobtaining an optimal billing structure and optimal load shifting profileuntil convergence is achieved; and d. controlling the energy consumptionof the user based on the optimal load shedding profile, optimal billingstructure, and optimal load shifting profile.
 2. The method of claim 1,wherein obtaining an optimal load shedding profile utilizes a knownbilling structure and a known load shifting profile for each user of thegrid.
 3. The method of claim 2, wherein obtaining an optimal loadshedding profile is performed without express information of loadpreferences of each user of the grid.
 4. The method of claim 1, whereineach of obtaining an optimal load shedding profile and obtaining anoptimal billing structure and an optimal load shifting profile is asolution to a robust convex optimization problem.
 5. A method for demandshaping through load shedding and load shifting in a electrical powersmart grid, comprising: a. directly controlling the load of a user totime shift part of the load demand of the user; b. providing energypurchase price incentives to the user to control the level of sheddingpart of the load demand of the user; and c. determining how much energyto purchase on each of the day-ahead market and the real-time market tocover the load demand of the user after load shifting and load shedding.6. The method of claim 5, wherein directly controlling the load of auser comprises calculating an optimal load shifting profile of the user.7. The method of claim 6, wherein calculating an optimal load shiftingprofile of the user comprises utilizing a robust optimization problemthat takes into account that a user load demand level and energypurchase prices are unknown stochastic processes.
 8. The method of claim5, wherein providing energy price incentives to the user comprisescalculating an optimal load shedding profile of the user.
 9. The methodof claim 8, wherein calculating an optimal load shedding profile of theuser comprises utilizing a robust optimization problem that takes intoaccount user load demand level and energy purchase prices are unknownstochastic processes.
 10. A method of controlling the energy demand of auser of an electrical power network, comprising: a. determining anoptimal amount of user energy demand to shed and a price incentiverequired to achieve said optimal amount; b. determining a billingstructure and an optimal amount of user energy demand to time-shift; c.repeating the first and second method steps until an equilibrium isachieved; and d. controlling the energy demand of the user based thedetermined optimal amount of user energy demand to shed, price incentiverequired to achieve said optimal amount, billing structure, and optimalamount of user energy demand to time-shift.
 11. The method of claim 10,wherein determining the optimal amount of user energy demand to shedcomprises formulating and solving an optimization problem using a knownbilling structure and a known user consumption for shiftable loads. 12.The method of claim 11, wherein formulating and solving comprisesutilizing standard convex optimization techniques to solve theoptimization problem in the case that energy price incentives are set bya party other than a network operator.
 13. The method of claim 11,wherein formulating and solving comprises setting a price offer for theuser and re-formulating and solving the re-formulated optimizationproblem with user preferences to load shedding in the case that energyprice incentives are set by a network operator.
 14. The method of claim10, wherein determining a billing structure and an optimal amount ofuser energy demand to time-shift comprises formulating and solving anoptimization problem based on results for calculating an optimal amountof user energy demand to shed and a price incentive required to achievesaid optimal amount.
 15. The method of claim 14, wherein formulating andsolving comprises formulating and solving respective billing structuresfor each of user shiftable plus sheddable load and user non-moveableload using a fixed utility ratio.
 16. The method of claim 15, whereinformulating and solving respective billing structures comprisere-formulating and solving re-formulated optimization problem to obtainan optimal amount of user energy demand to time-shift.
 17. A controlsystem for an electrical power network, comprising a controller thatshapes the electricity demand of users by determining electricitypurchase prices and billing structures in an iterative manner to achievean equilibrium and by determining the allocation of electricity demandfor each user given the determined electricity purchase prices andbilling structures.
 18. A smart electrical power network that provideselectricity to users and communicates data between its variouscomponents to control the flow of electricity, comprising a controlsystem that is adapted to shape user electricity demand bysimultaneously shifting and shedding user electricity demand and byprocuring electricity provided to users from a two-stage market with aday-ahead market (DAM) and a real-time market (RTM).